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Mon. Not. R. Astron. Soc. 000, 1–19 (0000) Printed 9 June 2016 (MN LATEX style file v2.2)

Cosmic Infrared Background anisotropies as a window into

primordial non-Gaussianity

Marco Tucci⋆, Vincent Desjacques† and Martin Kunz ‡

Departement de Physique Theorique and Center for Astroparticle Physics (CAP), University of Geneva,24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland

ABSTRACTThe angular power spectrum of the cosmic infrared background (CIB) is a sensitiveprobe of the local primordial bispectrum. CIB measurements are integrated over alarge volume so that the scale dependent bias from the primordial non-Gaussianityleaves a strong signal in the CIB power spectrum. Although galactic dust dominatesover the non-Gaussian CIB signal, it is possible to mitigate the dust contaminationwith enough frequency channels, especially if high frequencies such as the Planck 857GHz channel are available. We show that, in this case, measurements of the cosmicmicrowave background from future space missions should be able to probe the localbispectrum shape down to an amplitude |fNL| < 1.

Key words: cosmology: theory, large-scale structure of Universe, inflation

1 INTRODUCTION

Cosmology offers powerful probes of the high-energy physicsin the early Universe. One of the most important discrim-inants is primordial non-Gaussianity (PNG), which can inprinciple be tested across our entire past light-cone. Whilestandard single-field slow-roll inflationary models genericallypredict only very low levels of PNG, other scenarios can leadto a much larger non-Gaussianity in the primordial fluctua-tions (for more details see e.g. Bartolo et al. 2004; Komatsu2010; Chen 2010; Liguori et al. 2010; Yadav & Wandelt2010). Constraints or detections of PNG can thus give usinsight into the physics at energy scales that are otherwisevery difficult to access.

The Planck satellite (Planck Collaboration XVII 2015)has already put stringent constraints on primordial non-Gaussianity, but there is still much space for interesting phe-nomenology, especially if we can access the regime where theamplitude of the PNG is an order of magnitude smaller thanthe current limits.

While the cosmic microwave background (CMB) obser-vations are nearly cosmic-variance limited and, therefore,will not improve much over the Planck constraint, futuresurveys of the large scale structure (LSS) hold the promiseof achieving much tighter limits. Much effort has alreadybeen devoted to constrain PNG from a scale dependence inthe galaxy bias (Dalal et al. 2008; Matarrese & Verde 2008;

⋆ Email: Marco.Tucci@unige.ch† Email: Vincent.Desjacques@unige.ch‡ Email: Martin.Kunz@unige.ch

Slosar et al. 2008). Current LSS limits are at the level ofthe CMB pre–Planck constraints (Giannantonio et al. 2014;Leistedt et al. 2014), and they could improve by up to ∼ 1order of magnitude in the not so distant future (Agarwalet al. 2014; de Putter & Dore 2014; Raccanelli et al. 2015;Camera et al. 2015; Alonso & Ferreira 2015; Raccanelli et al.2015). Further improvements may come from higher-orderstatistics such as the galaxy bispectrum (see e.g. Scocci-marro et al. 2004; Sefusatti & Komatsu 2007; Jeong & Ko-matsu 2009), which encodes much more information on thePNG shape than the non-Gaussian bias. The latter is pre-dominantly sensitive to the ‘local’ bispectrum shape, whichpeaks on squeezed triangles.

Constraints on fNL from future measurements of thenon-Gaussian bias will strongly depend on the survey depthand volume (Carbone et al. 2010; Hamaus et al. 2011;de Putter & Dore 2014). Unfortunately, even the forthcom-ing large scale surveys such as Euclid (Laureijs et al. 2011)or LSST (Abell et al. 2009) will only cover a small frac-tion of the total comoving volume accessible to us. The cos-mic infrared background (CIB) traces the LSS over a muchlarger comoving volume and, thus, could potentially outper-form future galaxy redshift surveys. In this paper we showthat the CIB indeed is an excellent probe of primordial non-Gaussianity that allows us, in principle, to reach |fNL| < 1 inthe squeezed bispectrum limit. We demonstrate that such anerror can be achieved at Fisher matrix level even upon tak-ing into account the signal produced by galactic dust, whichis much stronger than the CIB at low multipoles. However,it is essential to have CIB maps available at enough frequen-cies.

c© 0000 RAS

2 Tucci et al.

The paper is organized as follows. We introduce ourhalo model of the CIB in §2, and present our forecast for adetection of the non-Gaussian bias in the CIB angular powerspectrum in §3. We discuss our findings and, in particular,the contamination by galactic dust in §4 and §5 before con-cluding in §6. Throughout the paper, we adopt the standardΛCDM cosmological model as measured by Planck (PlanckCollaboration XVI 2014): Ωm,ΩΛ,Ωbh

2, σ8, h, ns =0.3183, 0.6817, 0.02205, 0.8347, 0.6704, 0.9619.

2 MODEL OF THE COSMIC INFRAREDBACKGROUND

2.1 Halo model of the CIB intensity

In the usual Newtonian treatment, the observed CIB bright-ness is given by

I(ν, n) =

∫ χ∗

0

dz

(

dχ

dz

)

W (CIB)ν (z)

(

1 + δ)

(1)

where the integral runs over the (unperturbed) comovingdistance χ along the line of sight. It is typically computed upto redshift z∗ ∼ 10, corresponding to a comoving distanceχ∗ = χ(z∗), because of the negligible contribution of CIBfluctuations at higher redshifts. The redshift weight for CIBfluctuations is

W (CIB)ν (z) = a(z) ν(z) , (2)

where a(z) is the scale factor and ν(z) is the mean CIBemissivity per comoving volume at frequency ν:

ν(z) =

∫ Lcut

(1+z)ν

0

dL ng(L, z)L(1+z)ν

4π. (3)

Here, ng(L, z) denotes the infrared galaxy luminosity func-tion and L(1+z)ν is the infrared luminosity (in WHz−1) atthe rest–frame frequency (1 + z)ν. In practice, only faintsources below the flux detection limit Scut

ν are included inthe CIB, such that the integral in Eq. (3) is cut off at a(redshift and frequency dependent) luminosity Lcut

(1+z)ν(z).

The brighter sources are detected and removed from the skymaps. Finally, δ ≡ δ(ν, z, x = χ(z)n) is the perturbation tothe CIB emissivity at frequency ν, redshift z and comovingposition x = χ(z)n along the line of sight.

Let us focus first on the contribution from the averageemissivity. Following Shang et al. (2012), we split the meanemissivity into a sum of two contributions,

ν(z) =

∫

dM nh(M, z)[

fcν (M, z) + f s

ν(M, z)]

, (4)

where the average emissivity produced by the central andsatellite galaxies of a given halo at redshift z are

fcν (M, z) =

1

4πNcLc,(1+z)ν(M, z) (5)

f sν(M, z) =

1

4π

∫

dmns(m, z|M)Ls,(1+z)ν(m, z) . (6)

In the above expressions, nh(M, z) and ns(m,z|M) are thehalo and sub-halo mass functions, and M and m are theparent halo and sub-halo masses. The numbers Nc of cen-tral galaxies is specified by the halo occupation distribution(HOD; Berlind &Weinberg 2002; Zheng et al. 2005). Numer-ical simulations indicate that Nc typically follows a step-like

function (Kravtsov et al. 2004). We adopt a characteristicmass Mcen of the step function of 3 × 1011 M⊙/h (Zehaviet al. 2011; Desjacques et al. 2015), ignoring any luminos-ity dependence. In all subsequent calculations, we use theSO (Spherical Overdensity) halo and sub-halo mass func-tions provided by Tinker et al. (2008) and Tinker & Wetzel(2010), respectively, and integrate the sub-halo mass func-tion from a minimum halo mass Mmin = 1010 M⊙/h to theparent halo mass M . To identify halos, we adopt a densitythreshold of ∆c = 200 (in unit of the background densityρm(z)) at all redshifts.

In the model used here, the luminosity and clusteringof infrared galaxies are linked to the host halo mass. Thestrongly clustered galaxies are situated in more massive ha-los. They typically have more stellar mass and, therefore,are more luminous. Assuming the same luminosity-mass re-lation for both central and satellite galaxies, we relate thegalaxy infrared luminosity to the host halo mass throughthe parametric relation (Shang et al. 2012)

L(1+z)ν(M, z) = L0Φ(z)Σ(M)ΘCIB[(1 + z)ν] . (7)

As regards to the luminosity–mass relation, we shall followthe assumptions used in Planck Collaboration XXX (2014):

• L0 is an overall normalisation that is constrained frommeasurements of CIB power spectra. In principle, it shouldbe a constant but, in order to have a good fit to PlanckCollaboration XXX (2014), we allow small variations of L0

with the frequency, which are at most a 10 per cent between217 and 857GHz.

• The term Φ(z) describes the redshift-dependence of thenormalisation. We adopt a power-law scaling

Φ(z) = (1 + z)δ.

• We assume a log-normal distribution for the depen-dence Σ(M) of the galaxy luminosity on halo mass:

Σ(M) =M

√

2πσ2L/M

e−[log(M)−log(Meff)]

2/2σ2L/M , (8)

where Meff characterises the peak of the specific IR emissiv-ity and σM/L describes the range of halo masses that pro-duces a given luminosity L, which is fixed to σ2

M/L = 0.5.• For the galaxy spectral energy distribution (SED), we

assume a modified blackbody shape with a power-law emis-sivity as in Hall et al. (2010),

ΘCIB(ν) =

(ν/ν0)βBν(Td)/Bν0(Td) ν ≤ νb(ν/ν0)

−γ ν > νb, (9)

where Bν(T ) is the brightness of a blackbody with temper-ature T at the frequency ν, and ν0 is a reference frequency.The dust temperature Td is assumed to be a function ofredshift according to Td = T0(1 + z)α. The grey-body andpower-law connect smoothly at the frequency νb, at whichthe condition dlnΘ(ν, z)/dlnν = −γ is satisfied. For ourfiducial parameters, νb is always larger than 3000GHz, andhence the parameter γ has small relevance in the frequencyrange we are interested in.

For our fiducial halo model parameters, we adopt thevalues found in Planck Collaboration XXX (2014) by fitting(auto– and cross–) power spectra of the CIB anisotropiesmeasured by Planck and IRAS at 217, 353, 545, 857 and3000 GHz (see Table 1).

c© 0000 RAS, MNRAS 000, 1–19

CIB as a window into primordial non-Gaussianities 3

Table 1. Fiducial values of the parameters foud by Planck Col-laboration XXX (2014) for the luminosity-mass relation of thehalo model.

α T0 β γ δ log(Meff) L0

[K] [M⊙] [WHz−1]

0.36 24.4 1.75 1.7 3.6 12.6 0.95–1.05×10−3

We now turn to the fluctuations of the CIB brightnessacross the sky. In the Limber approximation, the angularpower spectrum of CIB fluctuations can be written as (e.g.,Knox et al. 2001; Song et al. 2003):

C(CIB)ℓ (ν, ν′) =

∫ z∗

0

dz

(

dχ

dz

)

1

χ2W (CIB)

ν (z)W(CIB)

ν′ (z)

× P ν×ν′

jj

(

k =ℓ

χ, z

)

. (10)

Here, P ν×ν′

jj is the cross-power spectrum⟨

δδ′

⟩

of fluctua-tions in emissivity at frequency ν and ν′. Pjj(k, z) can be setequal to the 3D power spectrum of galaxies, Pgg(k, z), un-der the assumption that fluctuations in the emissivity tracethose of the sources. In the context of the halo model (Scher-rer & Bertschinger 1991; Seljak 2000; Scoccimarro et al.2001; Cooray & Sheth 2002), galaxy power spectra are thesum of the contribution of the clustering in one single halo(1–halo term) and in two different halos (2–halo term):

P ν×ν′

gg (k, z) = P 1hgg (k, z, ν, ν

′) + P 2hgg (k, z, ν, ν

′) , (11)

where

P 1hgg (k, z, ν, ν

′) =1

jν jν′

∫

dMdN

dM

[

fcν (M, z)f s

ν′(M, z)u(k, z|M)

+ fcν′(M, z)f s

ν(M, z)u(k, z|M)

+ f sν(M, z)f s

ν′(M, z)u2(k, z|M)]

,

P 2hgg (k, z, ν, ν

′) =1

jν jν′

Dν(k, z)Dν′(k, z)Plin(k, z) , (12)

and

Dν(k, z) =

∫

dMdN

dMb1(M, z)u(k, z|M) (13)

×[

fcν (M, z) + f s

ν(M, z)]

.

Here, u(k, z|M) is the normalised Fourier transform of theNFW density profile (Navarro et al. 1997), Plin(k, z) is thelinear mass power spectrum extrapolated to redshift z, andb1 is the (Eulerian) linear halo bias (we use the fitting for-mula given in Tinker et al. 2010). In principle, Pjj(k, z) re-ceives also a shot noise contribution, but we will ignore it asit is typically of the same order as the 1-halo term.

At very large scales, i.e. at multipole ℓ <∼10, the Limberapproximation used in Eq. (10) is not valid. Assuming anegligible contribution from the 1–halo term, for angularscales ranging from ℓ = 2 to 40 CIB power spectra can be

computed by (Curto et al. 2015)

C(CIB)ℓ (ν, ν′) =

2

π

∫

dk k2 (14)

×∫ z∗

0

dz

(

dχ

dz

)

W (CIB)ν (z)jℓ(kχ(z))P

1/2gg (k, z, ν)

×∫ z∗

0

dz′(

dχ

dz

)

W(CIB)

ν′ (z′)jℓ(kχ(z′))P 1/2

gg (k, z′, ν′) ,

where jℓ(x) are spherical Bessel functions.

2.2 GR corrections and non-Gaussian bias

There are two important ingredients which we wish to addto the above model: General relativistic corrections and non-Gaussian bias. Details about the calculation of the GR ef-fects, which have already been considered in the context ofintensity mapping by Hall et al. (2013); Alonso et al. (2015),can be found in Appendix §A2.

At first order in GR perturbations, the observed CIBspecific intensity Eq. (1) eventually reduces to

I(ν, n) =

∫ χ∗

0

dz

(

dχ

dz

)

W (CIB)ν (z) (15)

×(

1 + δ +∂lnν∂η

δη + δ‖ + 2 s δ⊥

)

,

where δ is the perturbation to the galaxy emissivity in thesource rest-frame and at constant line-of-sight comoving dis-tance χ, the term proportional to δη arises from the transfor-mation to constant observed redshift, δ‖ is the perturbationto the source volume along the line-of-sight, and the termproportional to δ⊥ represents the perturbation to the lim-iting luminosity generated by fluctuations in the luminositydistance. Note that we have introduced a magnification bias

s(z) =∂lnν

∂Lcut(1+z)ν

=Lcut

(1+z)ν

4π

ng

ν(16)

in analogy with source number counts (Broadhurst et al.1995). The mean comoving emissivity ν is given by Eq. (3).In the conformal Newtonian gauge adopted for the calcula-tion, the perturbation to the conformal time η reads

Hδη = −Ψ−∫ χ

0

dχ′(

Ψ + Φ)

+ v · n (17)

while the perturbations to the source volume element par-allel and transverse to the beam are

δ‖ =

(

H− HH

)

δη +Ψ+ v · n (18)

+1

H

[

dΨ

dχ+(

Ψ + Φ)

− dv

dχ· n]

,

and

δ⊥ = Hδη − 1

χ

[

δη −∫ χ

0

dχ′(

Ψ+ Φ)

]

− Φ− κ , (19)

respectively. Here, overdots designate a partial derivativew.r.t. the unperturbed conformal time η. Note that χ and ηare related through χ = η0 − η. Furthermore,

κ =1

2

∫ χ

0

dχ′ χ− χ′

χχ′∇2

Ω

(

Ψ+Φ)

, (20)

where ∇2Ω is the Laplacian on the unit sphere, is the lensing

c© 0000 RAS, MNRAS 000, 1–19

4 Tucci et al.

convergence. We have restricted ourselves to adiabatic scalarperturbations. There would be more terms, should vector ortensor perturbations or isocurvature modes be present at anappreciable level.

Assuming that spatial variations in the comoving emis-sivity trace fluctuations in the galaxy number density, in-trinsic perturbations to the comoving emissivity δ in theNewtonian gauge are related to matter fluctuations δsyn

m ina synchronous gauge comoving with dark matter through(Challinor & Lewis 2011; Baldauf et al. 2011; Jeong et al.2012)

δ = b1 δsynm +

∂ ln ν∂η

v

k, (21)

where b1(k) = bG1 (k)+∆bNG1 (k) is the linear galaxy bias. We

have written b1(k) as generally the sum of a Gaussian piecebG1 (k) (which, in the halo model, depends on wavenumberthrough the profile u(k, z|M)) and a scale-dependent cor-rection ∆bNG

1 (k) induced by the primordial non-Gaussianity.Furthermore, v is a scalar function such that the (curl-free)velocity perturbation is v = k−1∇v in the Newtonian gauge.Finally, we have ignored the shot-noise contribution for thereasons stated above. Note that Eq. (21) is linear and, thus,corresponds to our halo bias b(M, z) in Eq. (13).

To relate the perturbations Ψ, Φ, δsynm and v to the

initial conditions, we use Φi(k) = (3/5)ζ(k), where ζ(k) isthe uniform-density gauge curvature perturbation on super-horizon scales (i.e. k ≪ H), as a reference. Our choice followsfrom the fact that, in the large scale structure literature, theprimordial non-Gaussianity is commonly laid down immedi-ately after matter-radiation equality (see, e.g., Desjacques &Seljak 2010, for a review), whence the factor of 3/5. The var-ious transfer functions will generally depend on the mattercontent of the Universe. Ignoring anisotropic stresses, theseare given by

TΨ(k, z) = g(z)T (k) (22)

TΦ(k, z) = TΨ(k, z)

Tδ(k, z) =2

3Ωm

(

k

H0

)2

T (k)D(z)

Tv(k, z) =

(Hk

)

f Tδ(k, z) .

Here and henceforth, g(z) and D(z) = a(z)g(z) are thegrowth function of potential and density perturbations, re-spectively, f is the logarithmic derivative f = dlnD/dlnaand the matter transfer function T (k) is obtained with theBoltzmann code CAMB in a synchronous gauge comovingwith the pressureless matter. Note that, with our defini-tions, the Fourier modes of the peculiar velocity v are givenby v(k, z) = ikHfTδ(k, z)Φi(k)/k

2.Non-Gaussianity generated outside the horizon induces

a 3-point function that is peaked on squeezed or collapsedtriangles for realistic values of the scalar spectral index. Theresulting non-Gaussianity depends only on the local valueof the curvature perturbation, and can thus be convenientlyparameterized by Φi = φ + fNL(φ

2 − 〈φ2〉), where φ de-notes a Gaussian field (Salopek & Bond 1990; Gangui et al.1994). The quadratic term introduces a coupling betweenshort- and long-wavelength modes, which results in a scale-

dependent halo bias at large scales (Dalal et al. 2008; Matar-rese & Verde 2008). This non-Gaussian bias takes the form(Slosar et al. 2008)

∆bNG1 (k, z) = 3fNL

(

∂lnnh

∂lnσ8

)

ΩmH20

a(z)TΦ(k, z)k2. (23)

It is this bias enhancement on large scales that we are tryingto measure with the CIB. A large, potentially detectablefNL & 1 can be produced e.g. by multiple scalar fields (Linde& Mukhanov 1997; Lyth et al. 2003).

We have assumed that the halo mass function is uni-versal and, thus, replaced the logarithmic derivative of themass function by δc(b1(M, z) − 1), where δc ∼ 1.68 is thepresent-day (linear) critical density threshold. For the lin-ear Gaussian bias, we use the fitting formula given in Tinkeret al. (2010).

Departure from Statistical Gaussianity in the initialconditions significantly affect the abundance of highly biasedtracers of the LSS, since their frequency sensitively dependson the tail of the density PDF (e.g. Lucchin & Matarrese1988). To ascertain the importance of this effect, we havereplaced the halo mass function in Eq. (4) by

nh(M, z, fNL) = nGh (M, z)

(

1 +R(M,z, fNL))

, (24)

where nGh (M, z) is our fiducial Gaussian, Tinker mass func-

tion and R(M, z, fNL) is the non-Gaussian fractional correc-tion modelled along the extensions proposed by Matarreseet al. (2000); Lo Verde et al. (2008) (see Desjacques & Seljak(2010) for a discussion). We have found that, for an inputvalue fNL = 1, including R(M,z, fNL) amounts to a ∼ 1%correction to the amplitude of the non-Gaussian CIB bias.We will thus ignore this effect in what follows.

2.3 Signature of PNG in the CIB anisotropies

The signature imprinted by the scale-dependent, non-Gaussian halo bias is largest at low multipoles where thecontribution of the 1-halo term is negligible. We have indeedchecked that, for the frequencies considered in our forecast,the 1-halo term is at most . 10% of the 2-halo contribu-tion. This corresponds to an effective nonlinear parameter|fNL| . 0.1, which is hardly accessible to CMB experimentswith the sensitivity considered here. Therefore, we can safelyneglect it at the low multipoles ℓ ≤ 40 where Eq. (14) isapplied. Notwithstanding, we include it at the higher mul-tipoles used to constrain the CIB model parameters since itdominates over the 2-halo term for ℓ & 1000.

At low multipoles ℓ . 40 where the Limber approxima-tion is not valid, we ignore the 1-halo term, but retain theGR corrections and the non-Gaussian bias so that the CIBangular power spectrum is computed as

C(CIB)ℓ (ν, ν′) =

2

π

∫

dk k2 (25)

×∫ z∗

0

dz

(

dχ

dz

)

W (CIB)ν (z)Fℓ(ν, k, χ(z))P

1/2Φi

(k)

×∫ z∗

0

dz′(

dχ

dz′

)

W(CIB)

ν′ (z′)Fℓ(ν, k, χ(z′))P

1/2Φi

(k) .

Here, PΦi(k) is the power spectrum of our reference curva-

ture pertubation, and is related to the linear matter power

c© 0000 RAS, MNRAS 000, 1–19

CIB as a window into primordial non-Gaussianities 5

Figure 1. GR corrections (solid thick lines) to the CIB power spectrum (solid thin lines) at 353 (left panel) and 857GHz (right panel).The different GR contributions are also isolated: the velocity term (dotted red lines); the redshift–space distorsion (short dashed bluelines); the ISW term (long dashed green term); the potential term (dot–dashed cyan lines). The GR corrections are relevant at the veryfirst multipoles, and become lower than 1 per cent at ℓ > 10. The dominant contribution comes from the velocity term.

Figure 2. CIB power spectra (including GR corrections) at 353 and 857GHz assuming Gaussian primordial fluctuations (black solidlines) and fNL = ±1 and ±5 (solid red and green lines). The two, upper and lower thick dotted lines indicate the cosmic variance

associated to the Gaussian case, whereas the median, thin dotted curve is the CIB power spectrum without the GR corrections. Alsoshown as blue dashed line is the dust contamination – reduced by a factor 100 – as measured by Planck Collaboration XXII (2015)for the cleanest sky patches with coverage fraction 10 (left line) and 40 per cent (right line). The dust contamination is still orders ofmagnitude larger than CIB fluctuations on large angular scales. At ℓ = 2 the CIB spectrum increases by about an order of magnitude aswe turn on fNL from 0 to ±5. At larger multipoles, the effect of non–Gaussianity quickly decreases, and it is lower than cosmic varianceat ℓ & 20 even for fNL ∼ 5. For negative values of fNL, the CIB power spectrum presents a minimum at ℓ <∼10 corresponding to theangular scale at which the NG bias contribution becomes dominant over the Gaussian bias term.

c© 0000 RAS, MNRAS 000, 1–19

6 Tucci et al.

Figure 3. The cosmic variance associated to the CIB powerspectrum with Gaussian initial conditions at 353GHz (red dot-ted line) is shown in comparison with the PNG signal ∆Cℓ =Cℓ(fNL) − Cℓ(0) for fNL = 1, 3 and 5 (black solid lines). Thered dashed line gives the cosmic variance reduced by a factor

√ℓ,

corresponding to power spectra binned in multipole intervals ofsize ℓ. The cosmic variance becomes lower than ∆Cℓ at ℓ <∼20 forfNL = 1 and lower or equal to ∆Cℓ up to ℓ>∼ 1000 for fNL = 3.We also plot the noise level for the Planck and COrE+ channelsat ν ∼ 350GHz (black dotted lines).

Figure 4. The redshift kernel for the CIB power spectrum atℓ = 2 for two different frequencies: 353GHz (black lines) and857GHz (blue lines). Dashed lines are for Gaussian primordialfluctuations, whereas solid lines assume local PNG with fNL =5. CIB anisotropies induced by PNG mainly arise from dustygalaxies at redshifts between 2.5 <∼z <∼6 with a mild dependenceon frequency.

spectrum through

Plin(k, z) =4

9Ω2m

(

k

H0

)4

T 2(k)D2(z)PΦi(k) . (26)

Furthermore, the radial window function Fℓ(ν, k, χ) is givenby

Fℓ(ν, k, z) = jℓ(kχ)

[

b1(ν, k, z)Tδ +∂lnν∂η

(

Tv

k− TΨ

H

)

+HH2

TΨ

+1

H(

TΨ + TΦ

)

]

+ j′ℓ(kχ)

[(

1

H∂lnν∂η

− HH2

+ 2

)

Tv

+k

HTΨ

]

− j′′ℓ (kχ)

(

k

H

)

Tv −(

1

H∂lnν∂η

− HH2

+ 1

)

×∫ χ

0

dχ′ jℓ(kχ′)(

TΨ + TΦ

)

, (27)

with the linear bias

b1(ν, k, z) =

∫

dM

[

bG1 (M, z) + 3fNLδc(

bG1 (M, z)− 1) ΩmH2

0

a TΦ k2

]

× nh(M, z)[

fcν (M, z) + f s

ν(M, z)]

u(k, z|M) (28)

predicted by the halo model and including the correction dueto primordial non-Gaussianity. Details about the calculationof Fℓ(ν, k, z) can be found in Appendix §A3. Note that Eq.(27) does not include the lensing magnification induced bythe luminosity cut–off, see Eq. (3). The typical flux densitycuts in CMB maps indeed are of the order of ≈ 100mJy,which correspond to luminosities > 1027WHz−1 at z > 1.These luminosities are much larger than the expected lumi-nosity of submillimeter galaxies.

In Fig. 1, the GR corrections to the CIB power spec-trum are shown at frequency 353 and 857GHz. As expected,these corrections are relevant only at the very large angularscales, ℓ <∼10. At ℓ = 2, they are of the same order of mag-nitude as the CIB spectrum (assuming Gaussian primordialfluctuations) and decrease to 1–2 per cent at ℓ = 10, withlittle dependence on the frequency. GR corrections are dom-inated by the velocity term, whereas the other terms can beconsidered negligible1.

The effect of primordial non–Gaussianity on the CIBpower spectrum is much more pronounced than the imprintof GR corrections so long as |fNL| >∼1. For |fNL| . 1, theGR corrections must be included in the analysis to avoidsignificant bias on fNL (see Camera et al. (2015) for relateddiscussion in the context of galaxy redshift surveys). In Fig-ure 2, we compare the CIB spectrum assuming fNL = ±5,±1 and 0, i.e. with primordial Gaussian conditions. GR cor-rections are also included (see the small increment in CIBspectra with fNL = 0 at the first ℓs). Like GR corrections,the non-Gaussian bias leads to a pronounced signal at lowmultipoles (ℓ <∼10), which decreases proportionally to ℓ−1.Namely, a local PNG with fNL ∼ 5 increases the CIB powerspectrum by one order of magnitude at ℓ = 2, and by a fac-tor of <∼2 at ℓ = 10. When fNL < 0, the CIB spectrum hastypically less power than in the Gaussian case. It reaches a

1 In Figure 1, the contribution to GR corrections from a singleterm are computed neglecting the other components. It should benoted that the total GR correction is not simply the sum of thesingle contributions because the double products among differentterms in Eq. (25) have to be included.

c© 0000 RAS, MNRAS 000, 1–19

CIB as a window into primordial non-Gaussianities 7

minimum at some multipole (due to a cancellation in the 2-halo term) before increasing again at very low ℓ. This leadsto a characteristic feature in the CIB power spectrum.

To gain some insight into the redshift and frequency de-pendence of the non-Gaussian signal, let us assume that theluminosity-mass relation can be approximated by a Diracdelta, i.e. Σ(M) ≈ δD(M − Meff). In this case, the non-Gaussian CIB bias ∆bNG

1 (ν, k, z) simplifies to

∆bNG1 (ν, k,z) ≈ 3

4πfNL

ΩmH20

aTΦ k2L0(1 + z)δΘCIB

[

(1 + z)ν]

× ∂

∂lnσ8

[

nh(Meff, z)ΘH(Meff −Mcen)

+

∫ ∞

Meff

dMnh(M, z)ns(Meff, z|M)

]

, (29)

where ΘH is the Heaviside step function. The integrand ofthe second term in the square bracket will be maximized forsome halo mass M = αMeff with α > 1 and, thus, can beapproximated as ≈ nh(αMeff, z)ns(Meff, z|αMeff). Note thatα will generally depend on redshift. A similar calculationyields

bG1 (ν, k, z) ≈L0

4π(1 + z)δΘCIB

[

(1 + z)ν]

(30)

×[

nh(Meff, z)bG1 (Meff, z)ΘH(Meff −Mcen)

+

∫ ∞

Meff

dMbG1 (M, z)nh(M, z)ns(Meff, z|M)

]

for the Gaussian part of the CIB bias. Hence, ignoring thecontribution from the satellite galaxies, the relative ampli-tude of the non-Gaussian CIB bias scales like

∆bNG1

bG1(ν, k, z) ∼ 3fNL

ΩmH20

a TΦ k2

(

bG1 (Meff, z))−1

(31)

× ∂lnnh

∂lnσ8(Meff, z)ΘH(Meff −Mcen) ,

i.e. it does not depend on ν. Therefore, the relative ampli-tude of the signal will weakly depend on the value of δ orthe exact shape of ΘCIB which, in our model, do not dependon halo mass. We thus expect that the CIB non-Gaussianbias mainly depends on the HOD parameters, i.e. Meff, Mcen

and the distribution of subhalos hosting galaxies. We will as-certain the sensitivity to HOD modeling in more detail in§5.

2.4 Galactic dust and cosmic variance

Figure 2 shows the two major constraints for the detectionof PNG in CIB spectra: the Galactic dust emission and thecosmic variance. The former dominates the CIB emissionat all the relevant frequencies by orders of magnitude, es-pecially on the largest angular scales. Its power spectrumis typically ∝ ℓ−2.4 and the amplitude is strongly depen-dent on the area of the sky considered. We will extensivelydiscuss this point later in §4. The second main constraintcomes from the cosmic variance. We can note that |fNL| >∼1is required in order to have corrections to CIB spectra largerthan the cosmic variance associated to the “standard” CIBspectrum. The signal is, in any case, tiny already at ℓ>∼ 20even for larger fNL and typically below the cosmic variance.However, cosmic variance can be significantly reduced by

combining information from independent multipoles. As ex-ample, if we assume to measure the CIB power spectra inbins of width ℓ, the cosmic variance will decrease by a fac-tor

√ℓ, as shown in Figure 3. The cosmic variance will be

then of the same level as or lower than the PNG signal for|fNL|>∼ 3 at all the multipoles considered.

It is well known that the redshit distribution of dustygalaxies which mainly contribute to CIB anisotropies slowlychanges with frequency. Namely, it moves to higher redshiftswhen the CIB is observed at longer wavelengths (e.g., PlanckCollaboration XVIII 2011). For example, we show in Fig-ure 4 the redshift integrand for the CIB spectra at ℓ = 2:the peak shifts from redshift 2 to ∼ 3 when the frequencychanges from 857 to 353GHz. In both cases, however, thecontribution from high–redshift galaxies (z >∼4) is not neg-ligible. It is interesting to consider the same in the presenceof PNG (in Figure 4, fNL = 5). At ℓ = 2, where the CIBspectrum is dominated by anisotropies induced by PNG,the integrand peaks even at higher redshifts, i.e. at z = 3.5–4.5 according to the frequency. CIB anisotropies induced byPNG are therefore mainly from dusty galaxies at redshiftssignificantly higher than for the “standard” CIB, covering aredshift interval between 2 <∼z <∼7.

3 FISHER MATRIX FORMALISM

We adopt a Fisher matrix approach to provide estimatesof the sensitivity of CIB measurements to primordial non–Gaussianity. In terms of power spectra, the Fisher matrixelement Fij can be written as (e.g., Tegmark et al. 2000)

Fij =

ℓmax∑

ℓ=ℓmin

2ℓ+ 1

2fsky Tr

(

C−1ℓ

∂Cℓ

∂θiC−1

ℓ

∂Cℓ

∂θj

)

, (32)

The model parameters θi and θj include the CIB modelparameters given in Table 1 plus the primordial non–Gaussianity parameter fNL and, when considered, the pa-rameters related to the dust emission. The covariance ma-trix Cℓ is an Nν × Nν matrix whose elements are definedas the auto– and cross–power spectra of data at Nν dif-ferent observational frequencies, i.e. (Cℓ)ij = C

νiνjℓ . In ab-

sence of residual foregrounds and CMB radiation, Cνiνjℓ =

C(CIB)ℓ (νi, νj)+Nℓ(νi)δij , where the diagonal elements of the

covariance matrix contain the Gaussian instrumental noiseterms:

Nℓ(ν) = w−1ν exp

(

ℓ(ℓ+ 1)θ2FWHM(ν)

8 log 2

)

, (33)

where w−1/2ν is the instrumental white noise level in

Jy sr−1/2 and θFWHM is the full-width at half-maximum beamsize in radians at the frequency ν. In Eq. (32), fsky is thefraction of the sky used to recover CIB fluctuations, and thefactor (2ℓ+1)fsky gives the effective number of uncorrelatedmodes per multipole.

The uncertainty on fNL is computed after marginalizingover the other parameters, i.e. upon inverting the Fisher ma-trix so that σ(fNL) =

√

[F−1]11 (where i = 1 in the Fishermatrix corresponds to the fNL parameter). In Eq. (32), wedefine the smallest observable multipole for an experimentto be ℓmin = π/(2f

1/2sky ), rounding up to the next integer.

c© 0000 RAS, MNRAS 000, 1–19

8 Tucci et al.

Table 2. Uncertainty on fNL obtained from a COrE+–like ex-periment in purely CIB+noise maps, using 40% of the sky, withdifferent frequency combinations and ℓmin = 3 and 20.

σ(fNL = 0)

# ν [GHz] ℓmin = 3 ℓmin = 20

1 340 0.68 3.02 340,520 0.52 1.44 [220, 520] 0.36 0.508 [220, 600] 0.21 0.22

As maximum multipole, we assume ℓmax = 1000. This guar-antees that the shot–noise contribution from star–formingdusty galaxies is negligible. Moreover, multipoles ℓ>∼ 1000provide negligible information on fNL.

As a first step, we apply the Fisher formalism to theideal case of CIB maps from which (Galactic) foregroundsand CMB have been perfectly removed. We refer to the in-strumental configuration of a possible future CMB missionlike COrE+2, as reported in Table 4. Hereafter, only frequen-cies higher than 200GHz will be considered in the analysis.Channels at lower frequencies are in fact dominated by theCMB and should be dedicated to removing CMB fluctua-tions from the signal.

This preliminarly test gives us an idea of the maximumlevel at which fNL can be detected through CIB observationsby future space missions dedicated to the measurement ofthe CMB polarization. In Table 2 we report the uncertaintyon fNL assuming fNL = 0, using 40% of the sky and a differ-ent number of frequencies. σ(fNL) is weakly dependent onthe value of fNL. We see that, in principle, |fNL| of 1–2 couldbe detectable with high significance (>∼ 3–σ) using 2–4 fre-quency channels, while |fNL| lower than 1 is accessible only ifNν > 4. We also quote the results obtained after neglectingthe first 20 multipoles, which are the most affected by Galac-tic dust residuals. While we observe a strong degradation inthe fNL sensitivity when only 1 or 2 frequency channels areemployed, the uncertainty on fNL only marginally increasesfor Nν ≥ 4. This proves that the very large angular scalesare not strictly required to detect the PNG signal.

We emphasize that an estimate of CIB fluctuations atdifferent frequencies is crucial to measure fNL. Apart fromgiving a better control of the foregrounds, it allows 1) a re-duction of the noise level by a factor ≈

√Nν (if all channels

have the same sensitivity) and 2) an improvement of thedetermination of CIB parameters (with σ(fNL) → 1/F11

for Nν > 2). In addition, because CIB anisotropies are notperfectly correlated at different frequencies, multi–frequencyCIB observations allow to combine signals from partly over-lapping volumes of the Universe. In the overlapping re-gions, these measurements trace similar matter fluctuations.Therefore, they can be combined to decrease cosmic vari-ance and improve the signal-to-noise ratio, analogously tothe multi-tracer technique in galaxy clustering (see e.g. Sel-jak 2009; Hamaus et al. 2011, for applications to fNL).

2 http://conservancy.umn.edu/handle/11299/169642

4 FORECASTS INCLUDING GALACTICDUST CONTAMINATION

Measuring CIB anisotropies is a challenging task, especiallyover large areas of the sky, even for future high–sensitivityCMB space missions. CIB fluctuations are a sub–dominantcomponent at all frequencies: CMB anisotropies dominateat frequencies ν < 200GHz (and over the CIB up to<∼350GHz), whereas Galactic dust emission is preponder-ant at higher frequencies. On the one hand, low–frequencytemplates are quite effective at subtracting the CMB com-ponent from maps at few hundreds of GHz (see, e.g. PlanckCollaboration XXX 2014, for a detailed discussion aboutthis point). On the other hand, distinguishing Galactic fromextragalactic dust emission is more difficult because of theirfairly similar spectral energy distribution (SED) which ap-proximately scales in both cases like a modified blackbodylaw. In Fig. 5, we plot the SED for the CIB intensity com-pared to the Galactic dust frequency spectrum (see belowfor the model description), normalized to the 100–GHz CIBtotal intensity. The spectra are almost identical at frequen-cies below 300GHz, but the CIB spectrum flattens at higherfrequencies, peaking at about 1000GHz. Therefore, the twoSEDs differ most in the Far–Infrared frequencies, suggestingthat observations at ≫ 500GHz will be crucial to separatethese two components.

On large areas of the sky, Galactic dust is orders ofmagnitude brighter than the CIB. Furthermore, it largelydominates the CIB at multipoles ℓ <∼200 in the cleanest re-gions of the sky (see Figure 2). Extracting the CIB signalfrom Galactic contamination thus requires a very accuratecomponent separation. Currently, typical component sepa-ration methods rely on NHI maps, which furnish a tracer ofthe dust gas (Planck Collaboration XXX 2014; Planck Col-laboration XVIII 2014). However, this method has only beenapplied successfully over limited areas of the sky, leaving al-ways Galactic dust residuals at the level of 5–10%. Theseresiduals strongly affects angular scales ℓ < 200. Compo-nent separation methods that exploit not only the frequencyspectral information but also the spatial information are nowbeing developed, with the aim to separate the two compo-nents over large areas of the sky in the Planck data (PlanckCollaboration XLVIII 2016).

4.1 Model of the Galactic thermal dust emission

Based on the latest Planck results (Planck Collaboration X2015), the frequency spectrum of the Galactic thermal dustcan be accurately described by a modified blackbody

Θd(ν) =

(

ν

ν0

)βd+3exp(γν0)− 1

exp(γν)− 1, (34)

where ν0 is the reference frequency (we choose ν0 =353GHz, if not otherwise specified) and γ = 2π~/kBTd. Theemissivity index βd and the dust temperature Td can varyon a pixel-by-pixel basis depending on the dust populationand environment. We fix them to the mean values found byPlanck (Planck Collaboration XXII 2015; Planck Collabo-ration X 2015), βd = 1.53 and Td = 19.6K.

The intensity of the dust emission strongly depends onthe area of the sky considered, with high–latitude regionsbeing noticeably less affected vy dust contamination. In

c© 0000 RAS, MNRAS 000, 1–19

CIB as a window into primordial non-Gaussianities 9

Figure 5. Frequency spectrum of the CIB brightness (solid line)and of the Galactic dust emission (dashed line) according to themodels used in the paper. Dust spectrum is normalized to the CIBbrightness at 100GHz. The observational frequencies of Planckand COrE+ are also shown.

Planck Collaboration XXII (2015), the dust angular powerspectrum at 353GHz has been measured at low multipoles,ℓ < 100, as a function of the Galactic mask (with fsky vary-ing from 40 to 80 per cent). The resulting power spectrum iswell fitted by a power law with a slope consistent with −2.4over all the Galactic masks. However, the amplitude variesnonlinearly as a function of fsky. Following their Eq. (D.3),for fsky between 0.4 and 0.8, we model the 353–GHz dustpower spectrum (in unit of Jy2 sr−1) as

Cdℓ (353) = Ad

(

ℓ

100

)αd

Ad(fsky) = 1.45 × 106(

fsky0.6

)[4.60+7.11ln(fsky/0.6)]

, (35)

with αd = −2.4. The amplitude of the spectrum decreasestherefore by a factor ∼ 7 from fsky = 0.8 to 0.6, and by afactor 2 from 0.6 to 0.4 (see Table 3).

Moving to smaller areas of the sky, we can get an esti-mate of the amplitude of dust power spectra from the em-pirical relation Cd

ℓ (ν) ∝ 〈Iν〉2, where 〈Iν〉 is the average dustintensity in the sky region considered. This relation, whichis somewhat expected, has been tested with data at differ-ent frequencies (Miville-Deschenes et al. 2007; Planck Col-laboration XXX 2016). Planck Collaboration XXX (2016)estimated 〈Iν〉 over large regions of the sky with fsky from0.24 to 0.72, and over 352 patches of 400 deg2 (fsky ≃ 0.01)at Galactic latitude |b| > 35. The dust mean intensity wasfound to lower from 0.167 to 0.106MJy sr−1 as fsky variesfrom 0.63 to 0.42. Based on the previous scaling relation,this translates into a decrease in the power spectrum ampli-tude of a factor 2.5, very close to the value given by Eq. (35)(i.e., 2.65). In the region with fsky = 0.24, the measuredmean intensity is 0.068MJy sr−1. This value is only about a

Table 3. Estimated amplitude of the dust angular power spec-trum at 353GHz as a function of the fraction of the sky.

fsky 0.8 0.6 0.4 0.2 0.1

Ad/106 [Jy2 sr−1] 9.78 1.45 0.72 0.29 0.12

factor 1.5–2 higher than the mean intensity in the cleanest400 deg2 patches (0.04–0.045 MJy sr−1). Using the measured〈Iν〉, we estimate Ad for fsky < 0.4, as reported in Table 3. Inparticular, the value of Ad for fsky = 0.1 has been obtainedassuming a sky fraction of 10% and a dust contaminationas low as the cleanest 400 deg2 patches observed by Planck.This is an indicative, albeit maybe optimistic, estimate.

It is convenient to factor the auto– and cross– dustpower spectra into a spatial term (Eq. 35), a frequency–dependence term (Eq. 34), and a frequency–correlation term:

Cdℓ (ν, ν

′) = Cdℓ (ν0)Θd(ν)Θd(ν

′)Rνν′ . (36)

The correlation between different frequency channels is thenencoded in the matrix R, which is assumed to be indepen-dent of the angular scales ℓ. If the signal is uncorrelated, Rreduces to the identity matrix (e.g., uncorrelated detectornoises) while, for perfectly correlated signals, it is a rank–1matrix containing only unit entries (e.g., CMB fluctuations).We expect that Galactic foregrounds typically fall into anintermediate case. Since we presently lack detailed measure-ments of the foreground correlation matrice R, we decidedto follow the simple model developed by Tegmark (1998);Tegmark et al. (2000) and write the correlation matrix interms of a parameter ζ, the frequency coherence:

Rνν′ ≃ exp

− 1

2

[

ln(ν/ν′)

ζ

]2

. (37)

The frequency coherence determines the extent to which twofrequencies can be separated before their correlation startsto break down. The two limits ζ → 0 and ζ → ∞ correspondto the two extreme cases discussed above. For foregroundswith a spectrum such as the dust emission, Tegmark (1998)showed that the frequency coherence is of the order of theinverse spectral index dispersion, ζ ≈ 1/

√2∆β, where ∆β

is the rms dispersion of the dust emissivity index. PlanckCollaboration XXII (2015) measured the mean value of β atintermediate latitudes for frequencies ≤ 353GHz and found〈β〉 = 1.51 with 1–σ dispersion of 0.07. We shall hereafteruse this value for ∆β, which leads to a frequency coherenceof ζ = 10.1. In the frequency range we are interested in, thisguarantees a very high level of correlation, always largerthan 99%. We will investigate in §5 the effects of havinglarger values of ∆β.

4.2 Dust foreground removal

Contamination by Galactic dust emission is the strongestlimitation for measurements of CIB fluctuations, and accu-rate methods to separate it from the CIB signal are required.In this section however, we are not interested in the abilityof a particular foreground subtraction method to separatedust from CIB. Instead, we shall hereafter assume that theGalactic dust removal can be done correctly down to a given

c© 0000 RAS, MNRAS 000, 1–19

10 Tucci et al.

level (e.g., 10%, 1%, etc.), so that we can investigate in whichway the presence of foreground residuals propagate into theuncertainty on the fNL parameter.

The problem is equivalent to forecasting the detectabil-ity of the CMB B–mode polarization and the tensor–to–scalar ratio r parameter in CMB experiments (see, e.g., Er-rard et al. 2016). Analogously to the CMB, CIB fluctuationscan be treated as being statistically isotropic on the sky, yetwith the important difference that the CIB frequency spec-trum is not perfectly known and depends on the CIB modelparameters.

Different authors (e.g., Tucci et al. 2005; Stivoli et al.2010; Errard et al. 2016) have shown that, after the subtrac-tion, power spectra of foreground contaminants leftover inCMB maps can be fairly well described by the original powerspectra scaled down by a factor that depends on details ofthe component separation and properties of signals. More-over, the noise variance in the reconstructed CMB mapswill be degraded according to the frequency spectrum offoregrounds and the noise in the channels involved in theforeground subtraction. Assuming that the frequency de-pendence of the foregrounds is perfectly known, the noisevariance in the reconstructed CMB maps – obtained by astandard minimum–variance solution – is (see, e.g., Tegmarket al. 2000; Stompor et al. 2009; Errard et al. 2016):

Σ2CMB =

[

(ATN−1A)−1

]

CMB CMB

, (38)

where A is the “mixing” matrix that describes the frequencydependence of the sky signal components (i.e., CMB andforegrounds), and has a dimension of [Nν × Ns] (Nν is thenumber of frequencies and Ns the number of sky compo-nents). The square [Nν ×Nν ] matrix N is the noise covari-ance matrix. Eq. (38) is a good approximation even whenthe spectral behavior of foregrounds is parametrized by aset of spectral parameters which need to be determined to-gether with the sky signal (see, however, the discussion inStompor et al. 2009).

We extend the previous formalism to the CIB, whichis now the component to be recovered. For simplicity, weassume that the sky signal is composed only of Galacticdust emission and CIB fluctuations. The elements Ai1 andAi2 of the [Nν×2] mixing matrix at the frequency i are thengiven by Eqs. (9) and (34), where the reference frequency ν0now is the frequency at which we want to recover the CIBmap. If the noise in the different channels is uncorrelated,the noise variance in CIB reconstructed maps is

Σ2CIB(ν0) =

∑

i Θ2d(i)/σ

2i

det(ATN−1A), (39)

with

det(ATN−1A) =∑

i

Θ2CIB(i)

σ2i

∑

i

Θ2d(i)

σ2i

(40)

−[

∑

i

ΘCIB(i)Θd(i)

σ2i

]2

,

and σi is the noise level at the ith frequency. We see that,in the regime ΘCIB ∼ Θd expected at frequencies <∼300GHz(see Figure 5), the noise in the reconstructed map diverges.Frequencies much larger than 300GHz are therefore manda-tory to separate CIB fluctuations and dust emission.

Following, e.g., Errard et al. (2016), we assume that themixing matrix A can be directly estimated with a set of fre-quency maps produced by one (or more) experiment. Thereconstructed CIB map (at our reference frequency ν0) willbe then obtained by linearly combining the set of frequencymaps on the basis of the mixing matrix. This can be seenas a general approach, independently of the specific compo-nent separation method employed. However, unlike for theCMB, we want to recover independent CIB maps at severalfrequencies. I f observations are available at n frequencies(ν >∼ 200GHz), we assume that “clean” CIB maps can be re-constructed only in nCIB ≤ n/2 frequencies, while the othernd = n − nCIB channels are dedicated to the estimation ofthe mixing matrix and the Galactic dust template used inthe subtraction. The noise variance in a recovered CIB mapis then given by Eq. (39), where the mixing matrix is evalu-ated using the nd “dust” channels plus the CIB channel atν0.

The auto– and cross– power spectra in clean CIB mapsread

Cνiνjℓ = C

(CIB)ℓ (νi, νj) + εCd

ℓ (νi, νj) + Σ2CIB(νi)δij , (41)

where ΣCIB is the noise variance obtained from Eq. (39), andε is the fraction of the total Galactic dust power spectrumleftover in CIB maps. We use as reference value ε = 10−2,which implies a subtraction of the dust emission at the levelof 99%. This level should be the minimum goal for futurespace missions characterized by high sensitivity and largefrequency coverage.

In the following forecasts, we will include two extra pa-rameters in the Fisher matrix (Eq. 32): the amplitude Ad ofthe dust power spectrum and the spectral power–law indexαd, shown Eq. (35). σ(fNL) will be also marginalized overthem.

4.3 Fisher forecasts for present and futureexperiments

In this section we aim to provide realistic forecasts for the de-tection of the fNL parameter including 1) the Galactic dustcontamination, as discussed above, and 2) the instrumentalproperties of CMB space missions. We focus on Planck, andon possible future experiments like COrE+, LiteBIRD (Mat-sumura et al. 2014) and PIXIE (Kogut et al. 2014). Table 4gives the instrumental specifications we consider for theseexperiments. We take into account frequencies ν > 200GHzsolely.

Planck. Observations from the Planck mission cover alarge range of frequencies, up to 857GHz. The 545– and 857–GHz channels are crucial for the CIB/dust separation. In ouranalysis we suppose that the channels at 353 and 857GHzare dedicated to provide the dust template, while CIB mapsare recovered at 217 and 545GHz. In Table 4 we report thedegraded sensitivity in these channels after the subtractionof the dust contamination (see Eq. 39). In this particularcase, the 217–GHz channel gains, in term of (squared) sen-sitivity, a factor ∼ 1.5, while the 545–GHz channel losessensitivity by almost a factor 6.

Assuming that the dust emission has been cleaned at alevel of 1%, we find that the uncertainty on fNL provided byPlanck is σ(fNL) ≃ 3.5, almost independently of the valueof fNL. In Figure 6 we show the relative uncertainty on fNL.

c© 0000 RAS, MNRAS 000, 1–19

CIB as a window into primordial non-Gaussianities 11

Figure 7. The value of fNL for which its relative uncertainty is equal to 1 (solid lines, corresponding to a 1σ detection), 1/2 (long dasedlines) and 1/3 (short dashed lines) is plotted as a function of the fraction of the dust residual. This is computed for Planck (left panel),COrE+ (middle panel) and the combination COrE+/Planck (right panel), in their best configuration for CIB measurements.

Figure 6. The relative uncertainty for positive fNL obtained byFisher forecasts for Planck (black lines), COrE+ (green lines)and a combination of COrE+ and Planck (red lines). The twoblack solid lines are for Planck with fsky = 0.2 (upper) and 0.4

(lower curve; the case with fsky = 0.6 is very close to the casewith 0.4); the dotted black line is for fsky = 0.1 and the dashedline (in the upper–right corner) for fsky = 0.8. For COrE+, thesolid (dashed) green line corresponds to the configuration with4 (3) CIB frequency maps. The red dashed line is for COrE+adding only the 857–GHz channels, and the solid line adding allthe Planck frequencies ≥ 353GHz (see the text and Table 4)

According to our estimates, Planck should be able to detect|fNL| at about 2–σ level if it is larger than 6, and give anupper limit of ∼ 3.5 at 1–σ. This would significantly im-prove the current constraints on the local fNL provided byPlanck using the CMB bispectrum (Planck CollaborationXVII 2015), i.e. fNL = 0.8 ± 5.0.

We have also considered CIB maps covering differentfractions of the sky. The optimal configurations have a skyfraction fsky ≃ 0.4, although similar results are obtained for

Figure 8. Signal–to–noise ratio for fNL = 5, 3 and 1 (solid,dashed and dotted lines, respectively) as expected for Planck (redlines), COrE+ (green lines) and the combination COrE+/Planck(red lines). This is computed for fsky = 0.4 and 1% dust residual.

0.2 <∼fsky <∼0.6. For smaller areas, the sensitivity to fNL getsworse, while very large fractions of the sky are not useful forthe fNL detection due to the strong dust contamination. Fi-nally, we show in Figure 7 the dependence of our results onthe amplitude of the dust residual. The possible detection offNL strongly depends on the dust contamination. In orderto improve the current Planck constraint on fNL, dust con-tamination has to be removed at least at a 3% level. On theother hand, a 1 permil cleaning would give an uncertaintyof σ(fNL) ≃ 2.

COrE+. This project represents an approximately op-timal space mission for the search of the CMB B–mode po-larization, characterized by a high sensitivity of a few µK–arcmin, a few arcmin resolution and a large frequency cover-age. The COrE+ project is planned to include 8 frequencies

c© 0000 RAS, MNRAS 000, 1–19

12 Tucci et al.

at ν > 200GHz, with the highest channel at 600GHz. Weconsider two possible configurations for the measurement ofthe CIB: a more conservative choice involving 3 frequenciesfor CIB and 5 for dust, and a optimistic one with half ofthe frequencies reserved for each component (see Table 4).In both cases, after the component separation, the noise inthe reconstructed CIB maps increases by a factor 5–10. Thissignificant degradation in sensitivity may be related to thefrequency coverage, which does not include the wavelengthsat which the SEDs of the two components start to differ.As expected however, the sensitivity on fNL significantlyincreases with respect to Planck, being σ(fNL)>∼ 1.6. Conse-quently, |fNL| = 5 would be measured at ∼ 3–σ, and |fNL|of 2–3 would be also detectable with a significance of about1– 2σ (see Figure 6). The difference between the two con-figurations is modest, although an extra CIB frequency hasbeen added. Here again, the best results are for sky fractionsaround 0.4–0.6. Looking at Figure 7, the COrE+ results ap-pear to be less dependent on the dust residual, presumablyowing to the higher instrumental sensitivity.

COrE+/Planck. The sensitivity of COrE+ to the fNL

parameter strongly improves when the highest Planck chan-nels are also taken into account and combined with COrE+data. Extending the frequency coverage to 857GHz is key tobetter separate CIB and dust emission. It is actually enoughto include the Planck 857–GHz channel in the mixing matrixto reduce the noise degradation in reconstructed CIB maps,at a level close to the COrE+ sensitivity (see Table 4). Thisis due to the different scaling of the dust and the CIB at highfrequencies, see Fig. 5. In Figure 6 we show the results whenPlanck data are included to estimate the mixing matrix andthe noise degradation. We see that a 3– (2–)σ detection of|fNL| is now possible down to values of 2 (1.5). This meansthat, with this configuration, the residual dust emission canbe accurately separated from the CIB fluctuations. This isconfirmed by the fact that reducing the level of the dustresidual there is a very small improvement in the detectionof fNL (see Figure 7). In addition, if we increase the fractionof the sky up to fsky = 0.8, σ(fNL) always decreases.

Figure 8 shows the signal–to–noise ratio (defined asSNR = fNL/σ(fNL)) for fNL as a function of the mini-mum observable multipole ℓmin for the three cases discussedabove. It is interesting to note that for Planck the SNR isconstant up to ℓmin = 20, indicating that at the largest angu-lar scales the signal is completely dominated by the dust con-tamination. On the contrary, COrE+ succeeds to better sep-arate the two components and to access information also atℓ <∼20. The multipole range that contributes the most to theSNR changes according to the experiment and the configu-ration: in Planck this range is about ℓ = 20–70; in COrE+ itextends almost up to ℓ = 300–400; in COrE+/Planck largeangular scales seem to contribute less than in the previousexamples, and the SNR clearly drops only at ℓ > 100 andthe main contribution comes from scales between ℓ ∼ 100and <∼600.

LiteBIRD. This satellite could provide interesting con-straints on fNL in its extended version3, with several spec-tral bands spanning 40–400GHz and 4 bands between 200and 400GHz. Clearly, without external information, Lite-

3 http://indico.cern.ch/event/506272/contributions/2138028/

BIRD alone is not suitable for detecting PNG through theCIB (σ(fNL)>∼ 6, see Table 4). However, jointly with Planck

data, we find σ(fNL) ≃ 1.3 (in a configuration where threeout of four LiteBIRD channels are dedicated to the CIB),only a factor 2 larger than with COrE+.

PIXIE. The planned PIXIE mission presents almostthe optimal requirements for detecting fNL through CIBanisotropies, thanks to its 400 channels between 30GHz and6THz and a rms sensitivity of the order of 70 nK in 1 pix-els. In these conditions, dust emission should be extremelywell constrained, even at the peak of the thermal emission.We expect therefore that PIXIE will be able to investigatevery tiny PNG signals, with |fNL| ≪ 1. As an example, wehave considered 8 frequency bands between 200 and 600GHzwith a squared sensitivity of 1 Jy sr−1 and a resolution of1.6: we obtain σ(fNL) ≃ 0.08 assuming 1% of dust residualand ∼ 0.07 without dust contamination. Extending the fre-quency range to 900GHz (but with a similar number of fre-quency bands) seems not to produce any relevant improve-ment in the fNL uncertainty. However, these results shouldbe taken as indicative. On one hand, we expect that in-creasing the number of frequency bands dedicated to CIB,σ(fNL) could still decrease. On the other hand, especiallyfor the PIXIE case that can investigate very low values offNL, a more accurate analysis should be required, taking intoaccount effects of other contaminants (CMB, extragalacticsources, etc.) and model uncertainties (e.g., in the CIB andbias prescriptions). We will discuss these issues in the nextSection.

5 DISCUSSION

Model of CIB power spectra. Our Fisher forecasts relyon the fact that the CIB model we adopt provides a gooddescription of actual CIB spectra, at least on the angularscales of interest. This is a reasonable hypothesis, consid-ering that the model is able to fit observations for auto–and cross–frequency power spectra, number counts, and ab-solute CIB levels simultaneously (Viero et al. 2013; PlanckCollaboration XXX 2014). However, although it successfullydescribes Planck data, it remains nonetheless a fairly simpli-fied description of the actual CIB. Future high–quality CIBmeasurements (necessary to detect the fNL parameter) mayrequire an additional level of sophistication.

The CIB angular power spectra are currently well deter-mined only at small/intermediate angular scales, i.e. ℓ>∼ 200.No information is thus far available at larger scales, wherethe PNG effects on CIB anisotropies are most pronounced.Some concern could arise if the shape of the low–ℓ part ofCIB spectra depends depends on model assumptions. Totest this, we have compared the large–scale power spectraobtained with different CIB models, all providing a good fitto the Planck data. Besides the extended halo model used inthis paper, we have considered: 1) a linear model where the2–halo contribution is determined by an effective bias and bythe star formation history, which in turn are constrained byPlanck data (see §5.4 of Planck Collaboration XXX 2014);2) a simplified halo model in which all galaxies have thesame luminosity regardless of their host dark matter haloused, e.g.. in Planck Collaboration XVIII (2011); Curto et al.(2015). Although they agree at ℓ>∼ 200, the power spectra

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CIB as a window into primordial non-Gaussianities 13

Table 4. Instrumental specifications of the space missions considered in the paper, the corresponding noise variance in the reconstructedCIB maps for different configurations (Σ2) and the uncertainty on fNL (assuming fNL = 0).

Planck σ(fNL)

ν [GHz] 217 353 545 857fwhm [arcmin] 5.02 4.94 4.83 4.64w−1 [Jy2 sr−1] 43.32 164.7 185.3 157.9

Σ2 [Jy2 sr−1] 29.0 – 1100 – 3.6

COrE+

ν [GHz] 220 255 295 340 390 450 520 600fwhm [arcmin] 3.82 3.29 2.85 2.47 2.15 1.87 1.62 1.40w−1 [Jy2 sr−1] 0.654 1.43 5.20 8.31 13.50 22.98 39.88 69.26

Σ2 [Jy2 sr−1] – 8.03 – 47.3 – – 377. – 1.8

– 8.04 – 47.3 – 212. 382. – 1.6

with Planck 857GHz – 1.83 – 10.0 – – 82.6 147. 0.7

with Planck – 1.97 – 10.8 – 45.6 90.2 163.6 0.6

LiteBIRD

ν [GHz] 235 280 337 402fwhm [arcmin] 30 30 30 30w−1 [Jy2 sr−1] 0.36 1.45 1.1 0.7

Σ2 [Jy2 sr−1] 11.3 149.5 6.4

with Planck 1.0 6.7 14.6 1.3

PIXIE

ν [GHz] 218 248 293 353 398 443 548 593fwhm [arcmin] 96 96 96 96 96 96 96 96

Σ2 [Jy2 sr−1] 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.08

no dust 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.07

exhibit a different behavior at the largest scales (see leftpanel of Figure 9). Our reference model peaks at higher an-gular scales and has significantly less power at ℓ <∼100. Themain factor that determines the low–ℓ behavior in the CIBspectra is the redshift evolution of the average galaxy emis-sivity, ν(z). In fact, when the same ν(z) is applied to thedifferent CIB models, the low–ℓ power spectra become verysimilar, and even indistinguishable (after a proper renor-malization) as shown in the right panel of Figure 9. Overall,the PNG features in the CIB power spectra seem to weaklydepend on the model prescription4.

In order to distinguish PNG signals, it is thereforekey to correctly model the evolution of the infrared galaxyemissivity. Accurate measurements of CIB power spectraat degree scales should further strengthen the model reli-ability, especially concerning this point. Estimates of num-ber counts, luminosity and correlation functions of infraredgalaxies at shorter wavelengths should also provide comple-mentary information. In addition, as shown in Planck Col-laboration XXX (2016), the evolution of the galaxy emissiv-ity can be related to the star formation density history. The

4 In Figure 9, for the linear model, we have considered only theGaussian case. In fact, for this model it is not straightforward tointroduce a NG bias due to the parametric approach used for theeffective bias of infrared galaxies (see Planck Collaboration XXX2014).

linear model and the extended halo model recover consistentstar formation histories below z = 2, in agreement with re-cent measurements by Spitzer and Herschel, while at higherredshifts, there are discrepancies between the two modelsand between models and observational estimates (see dis-cussion in Planck Collaboration XXX 2016). In the future,we expect that a better determination of the history of thestar formation density will improve significantly the con-straints on the galaxy emissivity and then will help to fixthe CIB spectral behavior at low ℓs.

Moreover, we have shown in §2.3 that the relative am-plitude of the non–Gaussian CIB bias could depend on theHOD parameter Meff, i.e. on the range of halo masses thatare most efficient at hosting star formation. In the extendedhalo model this parameter is assumed to be constant in time,in agreement with results from Behroozi et al. (2013), atleast out to z = 4. However, they found a much lower valuefor the characteristic halo mass (i.e. log(Meff/M⊙) = 11.7)compared to the Planck results. We have verified that, usingthis value for Meff, CIB spectra change in amplitude but notin shape, even with fNL 6= 0. This is an important indica-tion that the PNG signal is fairly insensitive to the HODparameters. Note, however, that we have extrapolated thesubhalo mass function of Tinker & Wetzel (2010) much be-yond its actual range of validity, i.e. 0.8 < z < 1.6. Further-more, while the amplitude non-Gaussian bias is generallygiven by derivative of nh relative to the normalisation σ8

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14 Tucci et al.

Figure 9. Left panel CIB power spectra with fNL = 0 and 5, ac-cording to different model approaches: the extended halo modelused in the paper (solid black lines); the simplified halo model(red long dashed lines); the linear model (green dash–dotted line,only fNL = 0). Short dotted lines shows the difference between

the Gaussian case and fNL = 5. Points represent Planck mea-surements (Planck Collaboration XXX 2016). CIB spectra differat ℓ < 200. Right panel The same as in the left panel, but herespectra are computed using the same galaxy emissivity evolutionas given by the extended halo model. Differences in CIB spectraare now small or negligible over all the multipole range consid-ered. In either case, the PNG signal seems to be independent onthe model approach.

(Slosar et al. 2008), we have assumed a universal halo massfunction nh, which implies ∂lnnh/∂lnσ8 = δcb1. The validityof this assumption may, however, depend on the halo iden-tification algorithm (Desjacques & Seljak 2010). AlthoughN-body simulations support universality when the mass ofSO halos is M & 1013M⊙ (Desjacques et al. 2009), the ex-tent to which this assumption holds at smaller masses is stilla matter of debate.

Galactic dust emission. Due to its relevance, weshould investigate the extent to which our results are sensi-tive to the Galactic dust model used in the analysis. Dustresidual in CIB reconstructed maps is parametrized – seeSection 4.1 and Eq. (36) – by the product of spatial, fre-quency and correlation terms. In particular, by taking thecorrelation matrix Rνν′ independent of the angular scale, weare supposing that the shape of dust residual power spec-tra are the same over all the frequencies. This is somethingexpected and supported by simulations of component sepa-ration performance (Curto et al. 2016).

Two free parameters are considered for dust residualpower spectra and then marginalized in the Fisher approach:the amplitude and the power–law index of the power spec-trum at the reference frequency. The dust SED is instead as-sumed to be perfectly known. This is motivated by the factthat this Galactic contaminant largely dominates the skyemission at ν >∼ 300GHz, especially on large angular scales,and the frequency spectrum is expected to be accurately de-

Table 5. Frequency coherence of the dust emission and the un-certainty on the fNL parameters as a function of the spatial dis-persion of the dust spectral index.

∆β 0.07 0.1 0.3 0.5

ν/ν′ Rνν′

1.5 > 0.99 > 0.99 0.985 0.962 > 0.99 > 0.99 0.96 0.893 > 0.99 0.99 0.90 0.744 0.99 0.98 0.84 0.62

σ(fNL)

Planck 3.6 4.2 7.5 9.1COrE+ 1.6 1.9 2.9 3.3

COrE+/Planck 0.6 0.64 0.72 0.76

termined across the sky. This hypothesis can be relaxed byadding as extra free parameter the dust emissivity index βd.We have verified however that it has a negligible impact onthe fNL uncertainty.

A more critical issue in the Fisher analysis is the fre-quency coherence in the dust residual. We have related itto the spatial dispersion of the dust emissivity index, ac-cording to the prescription proposed by Tegmark (1998).Planck Collaboration XXII (2015) found that the varianceacross the sky of the dust spectral index is ∆β ≃ 0.07 whichcorresponds to a frequency correlation level larger than 99%(between 200 and 900GHz). It is clear that reducing the fre-quency coherence of the dust emission can significantly de-grade the sensitivity on the fNL parameter for experimentscovering a large frequency range. In Table 5 we compute thefrequency coherence values as increasing ∆β. We can seethat dust emission is still highly correlated for ∆β ≃ 0.1,even between well separated frequencies, while some decor-relation is observed for ∆β ∼ 0.3 and ν/ν′ > 2. The un-certainty on fNL from Planck or COrE+ seems to be quitesensitive to the frequency coherence and increases by ∼ 20%if ∆β = 0.1. On the contrary, when Planck and COrE+ arecombined, the frequency decorrelation has little impact onresults.

Dust contamination and Fisher forecasts. As analternative approach to estimate fNL constraints in the pres-ence of dust contamination, we compute the Fisher matrixwithout assuming any foreground removal. We include in theanalysis all the observational frequencies for a given experi-ment at their nominal sensitivity. We expect this approachto produce more conservative results. Dust removal on mapsshould be in fact more powerful and effective than a CIB–dust emission separation at the level of power spectra. It isalso an useful test to check if our working hypothesis (e.g.,dust removal at a 1 per cent level in about half of the obser-vational frequencies) are too conservative or too optimistic.For Planck we find σ(fNL) ≃ 5, about a 50 per cent largerthan previous results with 1% dust residual. In this case, acomponent separation approach to remove the dust contam-ination is strictly required. On the other hand, in the case ofCOrE+, if we consider all the 8 frequencies between 200 and600GHz at their nominal sensitivity, the uncertainty on fNL

significantly decreases with respect to previous values, andwe find σ(fNL) ≃ 0.25 (to be compared with ∼ 1.6 and 0.6

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CIB as a window into primordial non-Gaussianities 15

for COrE+ and COrE+/Planck, respectively). This result isvery close to the one obtained without any dust contamina-tion in §3. Similarly, for LiteBIRD we obtain σ(fNL) ≃ 0.33.Such an improvement can be explained by the very smallnoise level expected in future experiments (not degraded bycomponent separation) together with a larger number of fre-quency channels. We therefore conclude that, in principle,the dust contamination should be fully controlled by high–sensitive observations at a sufficient number of frequencies.Previous results for COrE+ and LiteBIRD mission mightbe then conservative, at least in terms of the number of fre-quencies to be exploited for the CIB.

For the purpose of computing the Fisher matrix wefixed the fiducial cosmological model. This assumption con-tributes to a possible theoretical systematic uncertainty.However, the Planck satellite has measured the parame-ters of the cosmological standard model to high precision(Planck Collaboration XIII 2015). For example, the tiltof the primordial spectrum might be partially degeneratewith the CIB slope at low ℓ. But Planck has determinedthe value of the scalar spectral index with tiny error bars,ns = 0.968 ± 0.006. In addition, the Planck data covers awide range of scales, and there is no indication for a run-ning of the scalar spectral index or any other significantdeviation from a power law spectrum, except maybe a (notsignificant) power deficit on large scales (Planck Collabora-tion XX 2015). The power deficit can hardly be confusedwith the expected excess power on the largest scales in theCIB from non-Gaussian bias if fNL > 0, and also the curvesfor fNL < 0 turn back up at the lowest ℓ so that we do notexpect this to be a problem.

Finally, we did not fully take into account the non-Gaussianity of the CIB covariance matrix, although the CIBpower spectrum includes the non-Gaussian bias. We have ig-nored the contribution induced by a primordial four-pointfunction (or trispectrum) which, in the PNG model consid-ered here, is proportional to f2

NL. This term might not benegligible at low multipoles. The covariance will also includecontributions generated by the nonlinear gravitational evo-lution. They will involve, among others, the CIB bispectrum.However, these terms scale at best like k2 in the power of bi-ased tracers. Therefore, we expect them to contribute muchless than the 1-halo term at low multipoles.

Overall, σ(fNL) < 1 results should be taken with somecaution. For example, according to our estimates, PIXIEhas the potentiality to detect PNG signals from fNL < 0.1.However, as discussed in this section, our Fisher forecasts arebased on some theoretical assumptions (e.g., the bias pre-scription, Gaussian covariance matrix, etc.), and on a par-ticular CIB modeling framework, whose validity and impacton results has not been fully investigated. Finally, possiblesystematics in future data and extra astrophysical contam-inants (as CMB, point sources, thermal Sunyaev–Zeldovicheffect) have not been considered in the present analysis. Allthese aspects should be investigated in future works.

6 CONCLUSION

In this paper we investigate the ability of CIB observationsat frequencies of a few hundred GHz (i.e. where CMB exper-

iments operate) to detect local primordial non-Gaussianity,and measure the nonlinear parameter fNL by leveraging thescale dependent bias on large scales.

We model the CIB angular power spectrum under theassumption that the cosmic infrared background is pro-duced by star–forming galaxies emitting at infrared wave-lengths. We describe the clustering of these galaxies using ahalo model, with a galaxy luminosity–halo mass relation asadopted in Shang et al. (2012); Planck Collaboration XXX(2014). We take into account both general relativistic cor-rections and the scale–dependent non–Gaussian halo biasinduced by the local primordial non-Gaussianity. We per-form a Fisher forecast to ascertain the precision at whichfNL can be measured by CMB space missions.

We find that the GR corrections are subdominant ex-cept at the largest angular scales. The leading contributionarises from the Doppler term, although it only makes upless than 50% of the CIB angular power spectrum at 353GHz for ℓ = 2 (where cosmic variance is largest), higherfrequencies. At ℓ = 10, the GR corrections are reduced tothe percent level, corresponding to a local bispectrum shapewith amplitude fNL ≃ 0.1.

Under the ideal condition of perfect dust subtractionover 40% of the sky, future CMB space missions like COrE+,operating at frequencies between 200 and 600GHz, shall beable to detect fNL ∼ 1 with high significance (> 2 σ), whenseveral frequencies are combined. In this case, informationfrom the very large angular scales, i.e. the ones most affectedby Galactic dust, is not strictly required.

Of course, a complete subtraction of dust is unlikely.Therefore, we have also produced forecasts assuming a dustsubtraction at the 1% level in the best scenario. In this case,an experiment with a Planck–like sensitivity should be ableto reach an uncertainty on fNL of σ(fNL) ∼ 3.5 for skyfractions between 0.2 and 0.6. Larger sky fractions are notuseful because of the strong dust contamination, which ac-tually degrades the constraints. Future probes like COrE+,LiteBIRD and PIXIE, which have a higher sensitivity andmore frequency channels, should be able to constrain fNL

to much lower values, even in a situation where the lowestmultipoles could not be used owing to foreground contami-nation, systematic effects, or a residual dust fraction higherthan expected after the cleaning. We have found that thepresence of high-frequency channels in the range of 800 to1000 GHz is especially important to reduce the impact ofdust and reach |fNL| < 1. Future missions should thus ei-ther include detectors at those frequencies or,alternatively,combine their data with the Planck 857 GHz to achieve thisprecision.

To conclude, an analysis of current CIB observationsshould in principle already yield bounds on the local fNL

competitive with those obtained from the analysis of non-Gaussianity in the Planck CMB maps. The CIB holds thepromise of reaching |fNL| <∼1 in a not so distant future. Al-though not considered here, additional information couldalso be harvested from a cross-correlation of the “CMB lens-ing potential” and CIB anisotropies. This is expected to beless sensitive to the PNG but, at the same time, less affectedby dust contamination (Anthony Challinor, private commu-nication). While direct observations of the large scale struc-ture might eventually prove even more sensitive, the CIBwill remain an invaluable tool to cross-check detections or

c© 0000 RAS, MNRAS 000, 1–19

16 Tucci et al.

limits from LSS due to the different systematics present inthese various probes.

ACKNOWLEDGMENTS

It is a pleasure to thank Camille Bonvin and Anthony Challi-nor for helpful discussions. MT is also thankful to EnriqueMartınez–Gonzalez and Patricio Vielva for useful discus-sions about Fisher matrix forecasts. VD and MK are grate-ful to the Galileo Galilei Institute and the organizers of theworkshop on “Theoretical Cosmology in the Era of LargeSurveys”, during which part of this work was completed.The authors acknowledge support by the Swiss National Sci-ence Foundation. Part of the analysis was performed on theBAOBAB cluster at the University of Geneva.

APPENDIX A: GENERAL RELATIVISTICCORRECTIONS TO THE CIB INTENSITY

We provide details about the calculation of the GR cor-rection to the CIB brightness. We refer the reader to theintensity mapping studies of Hall et al. (2013); Alonso et al.(2015) for complementary information.

A1 CIB brightness

Let I(νo) be the observed CIB brightness (in units of Jysr−1), i.e. the amount of energy dEo that passes through aproper surface element dAo (transverse to the observed raydirection) per frequency interval dνo, per proper time dτoand per solid angle dΩo,

I(νo) =dEo

dτodνodΩodAo. (A1)

The corresponding number of photons received by the ob-server is dNo = dEo/hνo. Conservation of photon num-ber (which follows from Liouville’s phase space conserva-tion law) implies dNo = dNs, where d.Ns is the infinitesimalnumber of photons emitted by the source. Namely,

I(νo) =νoνs

dEs

dτsdνsdΩsdAs

dτsdνsdτodνo

dΩsdAs

dΩodAo(A2)

= (1 + z)−3 dEs

dτsdνsdΩsdAs.

To derive the second equality, we have used νo/νs =dτs/dτo = (1 + z)−1 and the reciprocity relation DL =(1 + z)DA, where DA =

√

dAs/dΩo and DL =√

dAo/dΩs

are the angular diameter distance and (redshift-corrected)luminosity distance, respectively. Note that the luminosityand angular distances have canceled each other out of thelast expression.

Let λ be the affine parameter of the light ray xµ(λ),so that kµ = dxµ/dλ is its wavevector. During an intervaldλ, the wavefront seeps out a proper volume dVs = dlsdAs,where dls = kµu

µs dλ is proportional to the 4-velocity uµ

s ofthe emitter. Therefore, the above relation becomes

I(νo) = (1 + z)−3 dEs

dτsdνsdΩsdVs(kµu

µs )dλ (A3)

= (1 + z)−3 physνs (kµuµs )dλ ,

where

physνs =dEs

dτsdνsdΩsdVs(A4)

is the source emissivity per physical volume (in unit of Jysr−1 m−1). In the case of a continuous medium, Eq. (A3)must be integrated along the light ray. Hence, we can write

I(νo, n) =

∫

dλ (kµuµs )

phys(1+z)νo(λ)

(1 + z)3, (A5)

where n is the observed angular position. In principle, weshould also take into account the loss of intensity as thephotons propagate through the intergalactic medium andmodify the above relation accordingly:

I(νo, n) =

∫

dλ (kµuµs )

phys(1+z)νo(λ)

(1 + z)3e−τ(νo,n,λ) , (A6)

where τ (νo, n, λ) is the optical depth along the path of theray. For simplicity however, we will ignore the beam absorp-tion in what follows, as the focus is the evaluation of Eq.(A5) at linear order in metric perturbations.

A2 GR corrections at first order in perturbations

We consider a perturbed FRW metric in the conformal New-tonian gauge with coordinates xµ = (η, xi),

ds2 = a2(η)(

ηµν + hµν

)

dxµdxν , (A7)

where

h00 = −2Ψ , h0i = 0 , hij = −2Φδij . (A8)

The perturbed photon wavevector kµ = dxµ/dλ correspond-ing to the light ray xµ(λ) can be written as

kµ =1

a2

(

− 1 + δν, n+ δn)

, (A9)

where δν is the fractional frequency perturbation and δn isthe change in the propagation direction, which is such thatn2 = 1 and n · δn = 0. Since lightlike geodesics are confor-

mally invariant, the metric (ηµν + hµν)dxµdxν admits the

same photon geodesics which we parametrize as xµ(λ). Thecorresponding photon wavevector kµ = dxµ/dλ is related tokµ through

kµ = a2kµ . (A10)

Hence, the two affine parameters satisfy λ = a2λ up to anirrelevant constant.

We calculate the change in the perturbed photonwavevector kµ using the geodesic equation

d

dλkµ = −Γµ

αβ kαkβ , (A11)

where Γµνλ are the Christoffel symbols in the conformally

transformed metric. At linear order in perturbations, onlythe first geodesic equation describing the frequency shiftis relevant as lensing of the uniform CIB yields the sameCIB. Nonetheless, lensing magnification induces a first or-der perturbation to the CIB brightness through the flux lim-iting value. We will discuss this effect below. The relevantChristoffel symbols are Γ0

00 = Ψ, Γ00i = Ψ,i and Γ0

ij = −Φδij ,

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CIB as a window into primordial non-Gaussianities 17

where X and X,i designate a derivative w.r.t. η and xi, re-spectively. Linearizing the first geodesic equation, we arriveat

d

dλδν = 2

dΨ

dλ+ Ψ + Φ (A12)

upon taking advantage of the fact that the derivative w.r.t.λ is

d

dλ≡ kµ∂µ ≈ −∂η + n

i∂i (A13)

at first order in perturbations. Eq. (A12) easily integratesto

δν(λ)− δνo = 2(

Ψ(λ)−Ψo

)

+

∫ λ

0

dλ′(Ψ + Φ)

. (A14)

It is now expressed in terms of the affine parameter λ, whichwe assume to be λ = 0 at the observer position. Further-more, Ψo = Ψ(λ = 0), δνo = δν(λ = 0) etc.

Assuming that the source at affine parameter λ (agalaxy contributing to the CIB) and the observer move witha 4-velocity uµ = a−1(1− Ψ, vi) and uµ

o = a−1o (1− Ψo, v

io),

where vi is the proper peculiar velocity, the redshift of theemitter relative to the observer (which could be measured ife.g. emission lines were resolved) is

1 + z(λ) =

(

a−2uµkµ)

λ(

a−2uµkµ)

o

≡ 1 + δz(λ)

a(

η(λ)) , (A15)

where 1/a(

η(λ))

is the source redshift in an unperturbeduniverse, and

δz(λ) = −(Ψ−Ψo)−∫ λ

0

dλ′ (Ψ + Φ)

+ (v− vo) · n (A16)

is the redshift perturbation. Note that the boundary termsat the observer position generate an unmeasurable monopole(−Ψo) and a dipole (vo · n), which we will ignore in whatfollows.

Since the emissivity physν explicitly depends on the ob-served redshift through ν = (1 + z)νo, it is convenient tore-parametrize the light ray as a function of z, so that

I(νo; n) =

∫

dz

(

kµuµ dλ

dz

)

phys(1+z)νo(z, x(z))

(1 + z)3. (A17)

Therefore, fluctuations are now defined at the hypersurfacesof constant observed redshift z. We begin by evaluating theterm kµu

µ(dλ/dz). Using kµuµ = a−1(η)(1 + z) with z(λ)

given by Eq. (A15), we obtain

kµuµ dλ

dz=

a2(η(z))

H(η(z))

(

1 + δν − a2(η(z))

H(η(z))

dδz

dλ

)

. (A18)

We must now take into account the fact that the coordinatetime fluctuates at fixed observed redshift z. Writing η(z) =η(z) + δη, where η is the conformal time corresponding tothe observed redshift z in the unperturbed background, theperturbation to a2/H reads

a2(η)

H(η)=

a2(z)

H(z)

[

1 +

(

2H− HH

)

δη

]

, (A19)

where a(z) ≡ a(η(z)), H(z) ≡ H(η(z)) and the overdot des-ignates a derivative w.r.t. η. The time perturbation δη canbe read off from Eqs. (A15) and (A16) as

Hδη = −Ψ−∫ λ(z)

0

dλ′(

Ψ + Φ)

+ v · n . (A20)

Here, λ(z) is the value of the affine parameter at the ob-served redshift z. Therefore, with the help of the relationHδη + δν = Ψ+ v · n, we eventually arrive at

kµuµ dλ

dz=

a2(z)

H(z)

(

1 + δ‖)

(A21)

where

δ‖ = −(

HH −H

)

δη +Ψ+ v · n (A22)

+a2(z)

H(z)

[

dΨ

dλ+(

Ψ + Φ)

− dv

dλ· n]

is the fractional change in dls, i.e. in the source volume ele-ment along the line of sight. Note that the second term in thesquare brackets can be evaluated using (d/dλ) = a2(d/dλ).

We now turn to the emissivity physν . With the time slic-ing adopted here, perturbations to the emissivity are definedat constant observed redshift z:

a3physν (z, x) = a3(z) physν (z)(

1 + δz (z, x))

, (A23)

where it is understood that all quantities are evaluated at afrequency ν = (1+z)νo. In practice, it is useful to relate the(gauge-invariant) fluctuation δz to the perturbation δN inthe conformal Newtonian gauge adopted for this calculation.δN is defined on slices of constant coordinate time η, whence

a3(z)physν (z)(

1 + δz (z, x))

= a3(η)physν (η)(

1 + δN (η, x))

(A24)Since, at fixed observed redshift, η(z) = η(z) + δη where δηis given by Eq. (A20), we find after some algebra

δz = δN +

(

˙physν

physν

+ 3H)

δη = δN +˙νν

δη (A25)

at first order in perturbations. Here and henceforth, ν =a3physν is the comoving emissivity. Taking into account thefactor (1 + z)−3 ≡ a3(z) in Eq. (A17), the CIB brightnesssimplifies to

I(νo; n) =

∫

dza2(z)

H(z)ν(z)

(

1 + δN +∂lnν∂η

δη + δ‖

)

,

(A26)with ν = (1 + z)νo.

Lastly, we must also take into account the fact thatthe CIB is produced by sources below a certain flux limitingvalue Scut

ν . Owing to fluctuations in the luminosity distance,this corresponds to a maximum luminosity Lcut

ν (z, n) whichdepends on redshift and position on the sky. At first orderin perturbations, we get

Lcutν (z, n) = Lcut

ν (z)(

1 + 2δ⊥)

. (A27)

The threshold luminosity Lcut(z) is related to the flux de-tection limit Scut through

Lcutν (z) = 4π

(

1 + z)4a2(z)χ2(z)Scut

ν , (A28)

where χ(z) is the line-of-sight comoving distance corre-sponding to the redshift z in the unperturbed background.The perturbation δ⊥ transverse to the photon propagationat the source position is given by Eq. (19). Therefore, theaverage CIB emissivity ν per comoving volume in Eq. (A26)

c© 0000 RAS, MNRAS 000, 1–19

18 Tucci et al.

should be replaced by

∫ Lcutν (z,n)

0

dL ng(L, z)L(1+z)ν

4π

= ν(z) + 2 ng

(

Lcutν (z)

) Lcutν (z)

4πδ⊥ ,

where, in ν(z), the integral over galaxy luminosities runsfrom 0 to Lcut

ν (z). The perturbed CIB comoving emissivityat hypersurfaces of constant observed redshift thus reads

ν(z, x) = ν(z)

(

1 + δN +∂lnν∂η

δη + δ‖ + 2ng

ν

Lcutν

4πδ⊥

)

(A29)for a given flux detection limit Scut

ν .Finally, all the background quantities are thus far been

evaluated at η(z). However, since the unperturbed photonpath can also be parametrized as xµ(χ) = (η0 − χ, χ), wecan recast the CIB brightness into Eq. (15) with the aid ofdχ/dz = a(η(z))/H(η(z)).

A3 Imprint on the CIB angular power spectrum

Fluctuations in the CIB intensity can be expanded in thebasis of spherical harmonics Y m

ℓ as

I(ν, n) =∑

ℓm

aℓm(ν)Y mℓ (n) . (A30)

The frequency-dependent multipoles aℓm(ν) are given by theline-of-sight integral

aℓm(ν) =

∫

dz

(

dχ

dz

)

a(z)ν(z)∆ℓm(ν, z) , (A31)

where χ now is the unperturbed line-of-sight comoving dis-tance (we drop hereafter the overline for shorthand conve-nience), and

∆ℓm(ν, z) =

∫

dnY m⋆ℓ (n)

(

δN +˙νν

δη + δ‖ + 2ng

ν

Lcut

4πδ⊥

)

=∑

S

∆Sℓm(ν, z) (A32)

has been schematically decomposed in §2.2 into a sum ofcontributions induced by the first-order Newtonian gaugeperturbations Ψ, Φ, v and the synchronous gauge densityδsynm . Going to Fourier space and using the plane-wave ex-pansion, some of the basic building blocks are, for instance,

∆Φℓm =

iℓ

2π2

∫

d3k TΦ(k, χ)jℓ(kχ)Ym⋆ℓ (k) Φi(k) (A33)

∆v·nℓm =

iℓ

2π2

∫

d3k Tv(k, χ)j′ℓ(kχ)Y

m⋆ℓ (k)Φi(k)

∆dvdη

·n

ℓm = − iℓ

2π2

∫

d3k k Tv(k, χ)j′′ℓ (kχ)Y

m⋆ℓ (k) Φi(k) .

In the second line, the prime denotes a derivative w.r.t. theargument of the spherical bessel function jℓ(x). Total deriva-tives w.r.t. the unperturbed conformal time, d/dη ≡ −d/dχ,bring an additional derivative w.r.t. the argument of jℓ(x).

Collecting all the terms and substituting into Eq. (A31),we eventually arrive at

aℓm(ν) =iℓ

2π2

∫

d3k

∫

dz

(

dχ

dz

)

a(z)ν(z) (A34)

× Fℓ(ν, k, z)Ym⋆ℓ (k)Φi(k) .

The radial window function Fℓ(ν, k, z) sourced by the per-turbations δN , δη and δ‖ is given by Eq. (27).

For the lensing magnification effect, which is not in-cluded in the Fisher analysis as it is small, Fℓ(ν, k, z) reads

Fℓ(ν, k,z) =

jℓ(kχ)

[(

1

Hχ− 1

)

TΨ − TΦ

]

+

(

1

Hχ− 1

)

×∫ χ

0

dχ′

[

(

TΨ + TΦ

)

+1

χ

(

TΨ + TΦ

)

]

jℓ(kχ′)

+1

2χ

∫ χ

0

dχ′ χ− χ′

χ′

(

TΨ + TΦ

)

jℓ(kχ′)

+ j′ℓ(kχ)

(

1− 1

Hχ

)

Tv

2 s(χ) , (A35)

with the magnification bias s(χ) given by Eq. (16).

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